A wavelet multiscale iterative regularization method for the parameter estimation problems of partial differential equations

A wavelet multi-scale method for the inverse problem of diffuse optical tomography

“…Indeed, because a subsampling of a factor 2 is required when applying Eqs. (24) and (25), Mallat's algorithm, Theorem 3, is designed for signal whose length is an integer power of two. When it is not the case, algorithms exist to apply the fast wavelet transform, e.g.…”

“…A relatively new method has emerged in the field of inversion, namely the wavelet multi-scale method [18][19][20][21][22][23][24][25]. This method relies on a reformulation of the original inverse problem into a sequence of sub-inverse problems of different scales using wavelet transform, from the largest scale to the smallest one.…”

“…This method relies on a reformulation of the original inverse problem into a sequence of sub-inverse problems of different scales using wavelet transform, from the largest scale to the smallest one. Successful applications of this method include the inversion of the Maxwell equations [19], the identification of space-dependent porosity in fluid-saturated porous media [20,21], the inversion of the two-dimensional acoustic wave equation [22], the reconstruction of permittivity distribution by the inversion of the electrical capacitance tomography model [23], the parameter estimation of elliptical partial differential equations [18,24] and the identification of space-dependent permeability in a nonlinear diffusion equation [25]. It is shown in these papers that the wavelet multi-scale method can enhance stability of inversion, accelerate convergence and cope with the presence of local minima to reach the global minimum.…”

Estimation of a Permeability Field within the Two‐Phase Porous Media Flow Using Nonlinear Multigrid Method

“…Wavelet multiscale method is a specific form of multiscale techniques, which has recently been used to solve various parameter estimation problems. Successful applications of this method include the Bayesian tomography [25,26], the Bayesian formulations of emission tomography [27], the thermal wave tomography [28], the diffuse optical tomography [29], the velocity estimation problems of a two-dimensional wave equation [30], the permeability estimation problems of a nonlinear convection-diffusion equation [31], and the parameter estimation problems of partial differential equations [32,33]. It is shown in these papers that the performance of iterative parameter identification methods is much better with the help of the multiscale techniques.…”